Analytical solutions of kinematic equations for erosion on a plane II. Rainfall of finite duration

This paper, part II of the series, derives analytical solutions to kinematic equations for erosion from a sloping plane subject to rainfall of finite duration. Both equilibrium and partial equilibrium cases are distinguished. Complete solutions of these cases have not appeared previously in hydrologic literature. Properties of these solutions are briefly discussed. Finally a procedure is suggested for estimating the parameters contained in these solutions.     

The meaning of oscillations in unit hydrograph S-curves

ABSTRACT

The Duhamel superposition integral is used to obtain some exact solutions for unit hydrograph applications. These equations and numerical examples are used to show that oscillations will occur in an S-curve when the time step is less than the excess rainfall duration if the measured hydrograph differs from a hydrograph that would be obtained by solving a linear differential equation with time-independent coefficients. The implications of this result with regard to the calculation of the instantaneous unit hydrograph (IUH) are discussed.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

Derivation of a curve number and kinematic-wave based flood frequency distribution

Abstract

The physically-based flood frequency models use readily available rainfall data and catchment characteristics to derive the flood frequency distribution. In the present study, a new physically-based flood frequency distribution has been developed. This model uses bivariate exponential distribution for rainfall intensity and duration, and the Soil Conservation Service-Curve Number (SCS-CN) method for deriving the probability density function (pdf) of effective rainfall. The effective rainfall-runoff model is based on kinematic-wave theory. The results of application of this derived model to three Indian basins indicate that the model is a useful alternative for estimating flood flow quantiles at ungauged sites.     

Incorporating uncertainty analysis into a regional IDF formula

A rainfall intensity–duration–frequency (IDF) relationship was generated by pooling annual maximum rainfall series from 14 recording rain gauges in southern Taiwan. Dimensionless frequency curves, plotted by the growth curve method, can be well fitted by regression equations for a duration ranging from 10 mins to 24 hours. As the parameters in regression equations have a good statistical relationship with average annual rainfall, a generalized regional IDF formula was then formulated. The formula, based on average annual rainfall as an index, can be easily applied to non-recording rain gauges. This paper further applies the mean value first-order second moment (MFOSM) method to estimate the uncertainty of the proposed regional IDF formula. From a stochastic viewpoint, the generalized regional IDF formula can accurately simulate the IDF relationship developed using frequency analysis (EV1) at individual stations. The method can provide both rainfall intensity and variance isohyetal maps for various rainfall durations and return periods over the study area. © 1998 John Wiley & Sons, Ltd.     

Diverging overland flow

This study investigates divering overland flow utilizing kinematic wave theory, which does not appear to have been dealt with previously. Explicit analytical solutions are derived in dimensionless form for space-time invariant rainfall. Analytical solutions do not seem to be tractable for time-varying rainfall. Depending upon the duration of rainfall, equilibrium and partial equilibrium cases are distinguished explicitly. The effect of divergence parameter on the hydrograph shape is shown. The adequacy of kinematic approximation for characterization of diverging overland flow is tested against laboratory watershed results. The diverging overland flow model is found to yield results which compare well with observations and with those of a plane model.     

Kinematic wave solutions for pollutant transport by runoff over an impervious plane,with instantaneous or finite‐period mixing

Kinematic wave solutions are derived for transport of a conservative non‐point‐source pollutant during a rainfall‐runoff event over an impervious plane. Rainfall is assumed to be steady, uniform and finite in duration. Prior to the start of rainfall, the pollutant is distributed uniformly over the plane. When rainfall occurs, the pollutant is washed off in a particular manner and the mixing of pollutant in the runoff water occurs either instantaneously or in a finite period of time under the assumption that the pollutant is soluble and is mixed completely in the runoff water. Copyright © 2002 John Wiley & Sons, Ltd.     

Kinematic wave solutions for pollutant transport over an infiltrating plane with finite‐period mixing and mixing zone

Kinematic wave solutions are derived for transport of a conservative non‐point‐source pollutant during a rainfall‐runoff event over an infiltrating plane for two cases: (i) finite‐period mixing and (ii) soil‐mixing zone. Rainfall is assumed to be steady, uniform and finite in duration, and it is assumed to have zero concentration of pollutants. Infiltration is assumed constant in time and space. Prior to the start of rainfall, the pollutant is distributed uniformly over the plane. In the first case, when rainfall occurs, the mixing of pollutant in the runoff water occurs in a finite period of time. In the second case, the chemical concentration is assumed to be a linearly decreasing function of rainfall intensity and overland flow. The solute concentration and discharge are found to depend on the flow characteristics as well as the solute concentration parameters. The characteristics of solute concentration and discharge graphs seem to be similar to those reported in the literature and observed in laboratory experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

Modelling catchment‐scale shallow landslide occurrence and sediment yield as a function of rainfall return period

A model‐based method is proposed for improving upon existing threshold relationships which define the rainfall conditions for triggering shallow landslides but do not allow the magnitude of landsliding (i.e. the number of landslides) to be determined. The SHETRAN catchment‐scale shallow landslide model is used to quantify the magnitude of landsliding as a function of rainfall return period, for focus sites of 180 and 45 km² in the Italian Southern Alps and the central Spanish Pyrenees. Rainfall events with intensities of different return period are generated for a range of durations (1‐day to 5‐day) and applied to the model to give the number of landslides triggered and the resulting sediment yield for each event. For a given event duration, simulated numbers of landslides become progressively less sensitive to return period as return period increases. Similarly, for an event of given return period, landslide magnitude becomes less sensitive to event duration as duration increases. The temporal distribution of rainfall within an event is shown to have a significant impact on the number of landslides and the timing of their occurrence. The contribution of shallow landsliding to catchment sediment yield is similarly quantified as a function of the rainfall characteristics. Rainfall intensity–duration curves are presented which define different levels of landsliding magnitude and which advance our predictive capability beyond, but are generally consistent with, published threshold curves. The magnitude curves are relevant to the development of guidelines for landslide hazard assessment and forecasting. Copyright © 2011 John Wiley & Sons, Ltd.     

Overland flow on converging and diverging surfaces — Assessment of numerical schemes

The similarity solutions of the equations of kinematic flow on a cone-shaped surface, provide a means for checking the numerical models used for converging and diverging flows. These models can be further improved if the initial excess rainfall curve can be described as a power of time. Illustration of this improvement and assessment of numerical models are given for a constant excess rainfall and for excess rainfall proportional to the inverse of the square root of time.     

Simple generalization approach for intensity–duration–frequency relationships

Rainfall intensity–duration–frequency (IDF) relationships describe rainfall intensity as a function of duration and return period, and they are significant for water resources planning, as well as for the design of hydraulic constructions. In this study, the two‐parameter lognormal (LN2) and Gumbel distributions are used as parent distribution functions. Derivation of the IDF relationship by this approach is quite simple, because it only requires an appropriate function of the mean of annual maximum rainfall intensity as a function of rainfall duration. It is shown that the monotonic temporal trend in the mean rainfall intensity can successfully be described by this parametric function which comprises a combination of the parameters of the quantile function a(T) and completely the duration function b(d) of the separable IDF relationship. In the case study of Aegean Region (Turkey), the IDF relationships derived through this simple generalization procedure (SGP) may produce IDF relationships as successfully as does the well‐known robust estimation procedure (REP), which is based on minimization of the nonparametric Kruskal–Wallis test statistic with respect to the parameters θ and η of the duration function. Because the approach proposed herein is based on lower‐order sample statistics, risks and uncertainties arising from sampling errors in higher‐order sample statistics were significantly reduced. The authors recommend to establish the separable IDF relationships by the SGP for a statistically favorable two‐parameter parent distribution, because it uses the same assumptions as the REP does, it maintains the observed temporal trend in the mean additionally, it is easy to handle analytically and requires considerably less computational effort. Copyright © 2012 John Wiley & Sons, Ltd.     

Rainfall characteristics for shallow landsliding in Seattle,Washington, USA

Shallow landsliding in the Seattle, Washington, area, has caused the occasional loss of human life and millions of dollars in damage to property. The effective management of the hazard requires an understanding of the rainfall conditions that result in landslides. We present an empirical approach to quantify the antecedent moisture conditions and rainstorm intensity and duration that have triggered shallow landsliding using 25 years of hourly rainfall data and a complementary record of landslide occurrence. Our approach combines a simple water balance to estimate the antecedent moisture conditions of hillslope materials and a rainfall intensity–duration threshold to identify periods when shallow landsliding can be expected. The water balance is calibrated with field‐monitoring data and combined with the rainfall intensity–duration threshold using a decision tree. Results are cast in terms of a hypothetical landslide warning system. Two widespread landslide events are correctly identified by the warning scheme; however, it is less accurate for more isolated landsliding. Copyright © 2005 John Wiley & Sons, Ltd.     

Design rainfall estimation in Australia: a case study using L moments and Generalized Least Squares Regression

Design rainfall is an important input to rainfall runoff models and is used for many other water resources planning and design applications. The estimation of design rainfall is generally done by applying a regional frequency analysis technique that uses data from a large number of rainfall stations in the region. This paper presents a regional rainfall frequency analysis technique that uses an L moments based index method coupled with Generalized Least Squares Regression (GLSR). The particular advantages of the GLSR method are that it accounts for the differences in record lengths across various sites in the region and inter-station correlation in deriving regional prediction equations. The proposed method has been applied to a data set consisting of 203 rainfall stations across Australia. It has been found that the proposed method can be applied successfully in deriving reasonably accurate design rainfall estimates from 1 to 72 h durations. It has also been found that the proposed method provides quite consistent estimates where a third order polynomial is adequate in smoothing the intensity–frequency–duration (IFD) curves. The method can readily be extended to a larger data set of Australia and other countries to derive generalized IFD data.     

An approach to analysing plot scale infiltration and runoff responses to rainfall of fluctuating intensity

Simulated rainfall of fluctuating intensity was applied to runoff plots on bare dryland soils in order to explore a new method for analysing the non‐steady‐state responses of infiltration and overland flow. The rainfall events all averaged 10 mm/h but included intensity bursts of up to 70 mm/h and lasting 5–15 min, as well as periods of low intensity and intermittency of up to 25 min. Results were compared with traditional steady‐state estimates of infiltrability made under simulated rainfall sustained at a fixed intensity of 10 mm/h. Mean event infiltration rate averaged 13.6% higher under fluctuating intensities, while runoff ratios averaged only 63% of those seen under constant intensity. In order to understand the changing soil infiltrability, up to three affine Horton infiltration equations were fitted to segments of each experiment. All equations had the same final infiltrability f_c, but adjusted values for coefficients f₀ (initial infiltrability) and K_f (exponential decay constant) were fitted for periods of rainfall that followed significant hiatuses in rainfall, during which subsurface redistribution allowed near‐surface soil suction to recover. According to the fitted Horton equations, soil infiltrability recovered by up 10–24 mm/h during intra‐event rainfall hiatuses of 15 to 20‐min duration, contributing to higher overall event infiltration rates and to reduced runoff ratios. The recovery of infiltrability also reduced the size of runoff peaks following periods of low intensity rainfall, compared with the predictions based on single Horton infiltration equations, and in some cases, no runoff at all was recorded from late intensity peaks. The principal finding of this study is that, using a set of affine equations, the intra‐event time variation of soil infiltrability can be tracked through multiple intensity bursts and hiatuses, despite the lack of steady‐state conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Continuous daily hydrograph simulation using duration curves of a precipitation index

The paper describes a parsimonious approach for generating continuous daily stream‐flow time‐series from observed daily rainfall data in a catchment. The key characteristic in the method is a duration curve. It is used to convert the daily rainfall information from source rain gauges into a continuous daily hydrograph at the destination river site. For each source rain gauge a time‐series of rainfall related ‘current precipitation index’ is generated and its duration curve is established. The current precipitation index reflects the current catchment wetness and is defined as a continuous function of precipitation, which accumulates on rainy days and exponentially decays during the periods of no rainfall. The process of rainfall‐to‐runoff conversion is based on the assumption that daily current precipitation index values at rainfall site(s) in a catchment and the destination site's daily flows correspond to similar probabilities on their respective duration curves. The method is tested in several small catchments in South Africa. The method is designed primarily for application at ungauged sites in data‐poor regions where the use of more complex and information consuming techniques of data generation may not be justified. Copyright © 2000 John Wiley & Sons, Ltd.     

Estimation of Nash's IUH parameters using stochastic search algorithms

In rainfall–runoff studies, it is often necessary to change the duration of a given unit hydrograph. Nash's Instantaneous Unit Hydrograph (IUH) is an ideal method that eliminates the hydrograph duration. This paper presents the results of the application of search algorithms, namely a genetic algorithm and hill climbing, to develop the IUH that minimizes the error between the observed and generated hydrographs. Also the performance of these methods has been compared with that of the classical method used for estimation of IUH, namely the method of moments. The genetic algorithm is a popular search procedure for function optimization that applies the mechanics of natural genetics and natural selection to explore a given search space. Hill climbing is an optimization technique that belongs to the family of local search and algorithms can be used to solve problems that have many solutions, with some solutions better than others. The results obtained from both the genetic algorithm and hill climbing algorithm for estimation of Nash's IUH parameters were compared with the results obtained by the method of moments for storms from two river basins that are located in different climatic regions. It was found that both the genetic algorithm and hill climbing provided improved and consistent results. Copyright © 2008 John Wiley & Sons, Ltd.     

Studies and evaluations on interception processes during rainfall based on a tank model

Some analyses are carried out with regard to canopy interception processes during rainfall events based on a tank model. A hypothesis, rainfall interception rate is proportional to the product of potential evaporation and rainfall intensity, is formed from past experimental data, and is applied to the data in this study. Computational equations are proposed to the interception rate and accumulative interception loss under constant rainfall intensity. Data from the Shirakawatani experimental forested catchment are used in order to examine the relationship between the interception rate and rainfall intensity, the ratio of the interception rate to rainfall intensity and potential evaporation, accumulative interception loss and the rainfall duration, and accumulative interception loss and accumulative rainfall. These regression relations show that interception processes are described by rainfall intensity and potential evaporation. An equation relating the aerodynamic resistance in the Penman–Monteith equation to rainfall intensity is proposed to explain the fact that the interception rate exceeds net radiation.     

Prediction of groundwater‐induced flooding in a chalk aquifer for future climate change scenarios

Current climate change models for the southeast UK predict changing rainfall patterns, with increased incidence of extreme events. The chalk aquifer in the UK and northern France is susceptible to groundwater‐induced flooding under such conditions. In this methodological study we apply a frequency domain analysis approach to the chalk aquifer to derive a transfer function between effective rainfall and groundwater level from 7 years of monitoring data from the North Heath Barn site, near Brighton. The derived transfer function was calibrated and validated against monitoring data and then used to predict groundwater level for rainfall models for high, medium and low emission scenarios from the UKCP09 database. The derived transfer function is most closely comparable to the linear aquifer model, despite evidence for both matrix and fracture or karst water flow in the chalk, with transmissivity and unconfined storativity at the catchment scale of 1548 m² day^?1 and 1.6 × 10^?2. The application of the transfer function to UKCP09 rainfall data suggests that groundwater‐induced flooding may be about four times more frequent by 2040–2069 compared with 1961–1990 and seven times more frequent by 2070–2099. The model data also suggest an increase in the duration of groundwater minima relative to the reference period. Compared to deterministic modelling which requires detailed knowledge of aquifer heterogeneity and processes, the transfer function approach, although with limitations, is simpler, incorporating these factors into the analysis through frequency and phase coefficients, and thus may have the potential for groundwater risk assessment in other areas. Copyright © 2015 John Wiley & Sons, Ltd.     

Rainfall stochastic disaggregation models: Calibration and validation of a multiplicative cascade model

The simulation of long time series of rainfall rates at short time steps remains an important issue for various applications in hydrology. Among the various types of simulation models, random multiplicative cascade models (RMC models) appear as an appealing solution which displays the advantages to be parameter parsimonious and linked to the multifractal theory. This paper deals with the calibration and validation of RMC models. More precisely, it discusses the limits of the scaling exponent function method often used to calibrate RMC models, and presents an hydrological validation of calibrated RMC models. A 8-year time series of 1-min rainfall rates is used for the calibration and the validation of the tested models. The paper is organized in three parts. In the first part, the scaling invariance properties of the studied rainfall series is shown using various methods (q-moments, PDMS, autocovariance structure) and a RMC model is calibrated on the basis of the rainfall data scaling exponent function. A detailed analysis of the obtained results reveals that the shape of the scaling exponent function, and hence the values of the calibrated parameters of the RMC model, are highly sensitive to sampling fluctuation and may also be biased. In the second part, the origin of the sensivity to sampling fluctuation and of the bias is studied in detail and a modified Jackknife estimator is tested to reduce the bias. Finally, two hydrological applications are proposed to validate two candidate RMC models: a canonical model based on a log-Poisson random generator, and a basic micro-canonical model based on a uniform random generator. It is tested in this third part if the models reproduce faithfully the statistical distribution of rainfall characteristics on which they have not been calibrated. The results obtained for two validation tests are relatively satisfactory but also show that the temporal structure of the measured rainfall time series at small time steps is not well reproduced by the two selected simple random cascade models.     

Basic Relations for Studying the Influence of Geophysical Processes on the Earth’s Rotation: The Angular Momentum Approach

Relations to study the influence of geophysical processes on the temporally varying rotation of the Earth are considered. Liouville’s equations of rotational motion are derived for a two-component Earth model (consisting of a solid mantle and a fluid core) and suitably simplified for calculations of the influence of mass redistributions on the Earth’s rotational behaviour. Excitation functions, or effective angular momentum functions, describing the influence of mass redistributions on the equations of rotational motion are derived, and their calculation is elucidated by some examples. Relations between temporally varying second degree Stokes coefficients of the gravity field and excitation functions are discussed. Different solutions of the equations of rotational motion are described. The identification of exciting geophysical processes by the kinematics of the inverse calculated excitation function is portrayed.